# Continuity 2

## One-sided continuity

- One-sided continuity is defined similarly to two-sided continuity.
- Let \(a\) be a number and let \(f(x)\) be a function.
- If we have \(\lim_{x\to a^-}f(x)=f(a)\), then we say that
*\(f(x)\) is continuous at \(a\) from the left*.
- If we have \(\lim_{x\to a^+}f(x)=f(a)\), then we say that
*\(f(x)\) is continuous at \(a\) from the right*.
- For example, consider the function \(f(x)=[[x]]\).
- If \(a\) is not an integer, then we have \(\lim_{x\to a^-}f(x)=[[a]]=\lim_{x\to a^+}f(x)\), therefore \(f(x)\) is continuous at \(a\).
- If \(a\) is an integer, then we have \(\lim_{x\to a^-}f(x)=[[a]]-1\) and \(\lim_{x\to a^+}f(x)=[[a]]\). Therefore, \(f(x)\) is continuous at \(a\) from the right.
- Exercise. 2.5.42.

## Continuity on an interval

- Let \(f(x)\) be a function and \(I\) an interval. Then
*\(f(x)\) is continuous on \(I\)*, if it is continuous at each point of \(I\).
- If an endpoint \(a\) of \(I\) is closed, and \(f(x)\) is only defined on the side of \(a\) in the direction of \(I\), then instead of requiring two-sided continuity at \(a\), we only require that \(f(x)\) is continuous at \(a\) from the direction of \(I\).
- For example, let \(f(x)=1-\sqrt{1-x^2}\) and \(I=[-1,1]\).
- For \(-1<a<1\), we require two-sided continuity: \[
\lim_{x\to a}f(x)=\lim_{x\to a}(1-\sqrt{1-x^2})=1-\lim_{x\to a}\sqrt{1-x^2}=1-\sqrt{1-a^2}=f(a).
\]
- Since the domain of \(f(x)=1-\sqrt{1-x^2}\) is \([-1,1]=I\), in case \(a=\pm1\), we only require continuity from the direction of \(I\): \[
\lim_{x\to(-1)^+}f(x)=1=f(-1),\quad\lim_{x\to1^-}f(x)=1=f(1).
\]
- All this shows that \(f(x)\) is continuous on \(I\).
- Click here for a plot.
- Note that the graph of \(f(x)\) is the lower half of the circle \[
x^2+(y-1)^2=1.
\]
- We say that
*\(f(x)\) is continuous everywhere*, if it is continuous on the entire real line \(\mathbf R=(-\infty,\infty)\).
- Exercises. 2.5: 26,46